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ELEMENTARY MATHEMATICS

W W L CHEN and X T DUONG

c© W W L Chen, X T Duong and Macquarie University, 1999.

This work is available free, in the hope that it will be useful.

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Chapter 4

INDICES AND LOGARITHMS

4.1. Indices

Given any non-zero real number a ∈ R and any natural number k ∈ N, we often write

ak = a × . . . × a︸︷︷︸k

. (1)

This definition can be extended to all integers k ∈ Z by writing

a0 = 1 (2)

and

ak =1

a−k=

1a × . . . × a︸︷︷︸

−k

(3)

whenever k is a negative integer, noting that −k ∈ N in this case.It is not too difficult to establish the following.

LAWS OF INTEGER INDICES. Suppose that a, b ∈ R are non-zero. Then for every m, n ∈ Z,we have(a) aman = am+n;

(b)am

an= am−n;

(c) (am)n = amn; and(d) (ab)m = ambm.

† This chapter was written at Macquarie University in 1999.

4–2 W W L Chen and X T Duong : Elementary Mathematics

We now further extend the definition of ak to all rational numbers k ∈ Q. To do this, we first of allneed to discuss the q-th roots of a positive real number a, where q ∈ N. This is an extension of the ideaof square roots discussed in Section 1.2. We recall the following definition, slightly modified here.

Definition. Suppose that a ∈ R is positive. We say that x > 0 is the positive square root of a ifx2 = a. In this case, we write x =

√a.

We now make the following natural extension.

Definition. Suppose that a ∈ R is positive and q ∈ N. We say that x > 0 is the positive q-th root ofa if xq = a. In this case, we write x = q

√a = a1/q.

Recall now that every rational number k ∈ Q can be written in the form k = p/q, where p ∈ Z andq ∈ N. We may, if we wish, assume that p/q is in lowest terms, where p and q have no common factors.For any positive real number a ∈ R, we can now define ak by writing

ak = ap/q = (a1/q)p = (ap)1/q. (4)

In other words, we first of all calculate the positive q-th root of a, and then take the p-th power of thisq-th root. Alternatively, we can first of all take the p-th power of a, and then calculate the positive q-throot of this p-th power.

We can establish the following generalization of the Laws of integer indices.

LAWS OF INDICES. Suppose that a, b ∈ R are positive. Then for every m, n ∈ Q, we have(a) aman = am+n;

(b)am

an= am−n;

(c) (am)n = amn; and(d) (ab)m = ambm.

Remarks. (1) Note that we have to make the restriction that the real numbers a and b are positive.If a = 0, then ak is clearly not defined when k is a negative integer. If a < 0, then we will have problemstaking square roots.

(2) It is possible to define cube roots of a negative real number a. It is a real number x satisfyingthe requirement x3 = a. Note that x < 0 in this case. A similar argument applies to q-th roots whenq ∈ N is odd. However, if q ∈ N is even, then xq ≥ 0 for every x ∈ R, and so xq �= a for any negativea ∈ R. Hence a negative real number does not have real q-th roots for any even q ∈ N.

Example 4.1.1. We have 24 × 23 = 16 × 8 = 128 = 27 = 24+3 and 2−3 = 1/8.

Example 4.1.2. We have 82/3 = (81/3)2 = 22 = 4. Alternatively, we have 82/3 = (82)1/3 = 641/3 = 4.

Example 4.1.3. To show that√

5 + 2√

6 =√

2+√

3, we first of all observe that both sides are positive,and so it suffices to show that the squares of the two sides are equal. Note now that

(√

2 +√

3)2 = (√

2)2 + 2√

2√

3 + (√

3)2 = 5 + 2 × 21/231/2 = 5 + 2(2 × 3)1/2 = 5 + 2√

6,

the square of the left hand side.

Example 4.1.4. To show that√

8 − 4√

3 =√

6 −√

2, it suffices to show that the squares of the twosides are equal. Note now that

(√

6 −√

2)2 = (√

6)2 − 2√

6√

2 + (√

2)2 = 8 − 2 × 61/221/2 = 8 − 2(6 × 2)1/2

= 8 − 2√

12 = 8 − 2 × 2√

3 = 8 − 4√

3,

the square of the left hand side.

Chapter 4 : Indices and Logarithms 4–3

In the remaining examples in this section, the variables x and y both represent positive real numbers.

Example 4.1.5. We have

x−1 × 2x1/2 = 2x(1/2)−1 = 2x−1/2 =2

x1/2=

2√x

.


(27x2)1/3 ÷ 13(x5)1/2 = 271/3(x2)1/3 ÷ 1

3x5×(1/2) = 3x2/3 ÷ x5/2

3= 3x2/3 × 3

x5/2

= 3x2/3 × 3x−5/2 = 9x2/3x−5/2 = 9x(2/3)−(5/2) = 9x−11/6 =9

x11/6.


(9x)1/2(8x−1/2)1/3 = 91/2x1/2 × 81/3x(−1/2)×(1/3) = 3x1/2 × 2x−1/6 = 6x(1/2)−(1/6) = 6x1/3.


(8x3/4)−2 ÷(

12x−1

)2

= (8x3/4)−2 ×(

12x−1

)−2

= 8−2x(3/4)×(−2) ×(

12

)−2

x(−1)×(−2)

=182

x−3/2 × 22x2 =116

x(−3/2)+2 =116

x1/2 =√

x

16.


3√

8a2b × a1/3b5/3 = (8a2b)1/3a1/3b5/3 = 81/3(a2)1/3b1/3a1/3b5/3 = 2a2/3b1/3a1/3b5/3

= 2a(2/3)+(1/3)b(1/3)+(5/3) = 2ab2.


4√

(16x1/6y2)3 = ((16x1/6y2)3)1/4 = (16x1/6y2)3/4 = 163/4(x1/6)3/4(y2)3/4

= (161/4)3x(1/6)×(3/4)y2×(3/4) = 8x1/8y3/2.

We now miss out some intermediate steps in the examples below. The reader is advised to fill in allthe details in each step.


16(x2y3)1/2

(2x1/2y)3× (4x6y4)1/2

(6x3y1/2)2=

16xy3/2

8x3/2y3× 2x3y2

36x6y=

16xy3/2 × 2x3y2

8x3/2y3 × 36x6y=

32x4y7/2

288x15/2y4

=32288

x4y7/2x−15/2y−4 =19x−7/2y−1/2 =

19x7/2y1/2

.


(3x)3y2

(5xy)2÷ (5xy)4

(27x9)3=

(3x)3y2

(5xy)2× (27x9)3

(5xy)4=

(3x)3y2 × (27x9)3

(5xy)2 × (5xy)4=

33273x30y2

56x6y6

=53144115625

x30y2x−6y−6 =53144115625

x24y−4 =531441x24

15625y4.



x−1 + y−1

x + y− x−1 − y−1

x − y=

(x − y)(x−1 + y−1) − (x−1 − y−1)(x + y)(x − y)(x + y)

=(1 + xy−1 − yx−1 − 1) − (1 + yx−1 − xy−1 − 1)

(x − y)(x + y)=

2(xy−1 − yx−1)(x − y)(x + y)

=2

(x − y)(x + y)×

(x

y− y

x

)=

2(x − y)(x + y)

× x2 − y2

xy=

2xy

.


x−1 − y−1

x−2 − y−2= (x−1 − y−1) ÷ (x−2 − y−2) =

(1x− 1

y

)÷

(1x2

− 1y2

)=

y − x

xy÷ y2 − x2

x2y2

=y − x

xy× x2y2

y2 − x2=

x2y2(y − x)xy(y2 − x2)

=x2y2(y − x)

xy(y − x)(y + x)=

xy

x + y.

Alternatively, write a = x−1 and b = y−1. Then

x−1 − y−1

x−2 − y−2=

a − b

a2 − b2=

1a + b

= (a + b)−1 =(

1x

+1y

)−1

=(

x + y

xy

)−1

=xy

x + y.


x2y−1 ÷ (x−1 + y−1) =x2

y÷

(1x

+1y

)=

x2

y÷ x + y

xy=

x2

y× xy

x + y=

x3

x + y.


xy ÷ ((x−1 + y)−1)−1 = xy × (x−1 + y)−1 = xy ×(

1x

+ y

)−1

= xy ×(

1 + xy

x

)−1

= xy × x

1 + xy=

x2y

1 + xy.

4.2. The Exponential Functions

Suppose that a ∈ R is a positive real number. We have shown in Section 4.1 that we can use (1)–(4)to define ak for every rational number k ∈ Q. Here we shall briefly discuss how we may further extendthe definition of ak to a function ax defined for every real number x ∈ R. A thorough treatment of thisextension will require the study of the theory of continuous functions as well as the well known resultthat the rational numbers are “dense” among the real numbers, and is beyond the scope of this set ofnotes. We shall instead confine our discussion here to a heuristic treatment.

We all know simple functions like y = x2 (a parabola) or y = 2x + 3 (a straight line). It is possibleto draw the graph of such a function in one single stroke, without lifting our pen from the paper beforecompleting the drawing. Such functions are called continuous functions.

Suppose now that a positive real number a ∈ R has been chosen and fixed. Note that we havealready defined ak for every rational number k ∈ Q. We now draw the graph of a continuous functionon the xy-plane which will pass through every point (k, ak) where k ∈ Q. It turns out that such afunction is unique. In other words, there is one and only one continuous function whose graph on thexy-plane will pass through every point (k, ak) where k ∈ Q. We call this function the exponentialfunction corresponding to the positive real number a, and write f(x) = ax.

x1 2-2 -1

1

4

3

y

1/21/4

2

x1 2-2 -1

2

4

6

y

9

7

5

3

1

y

8


LAWS FOR EXPONENTIAL FUNCTIONS. Suppose that a ∈ R is positive. Then a0 = 1. Forevery x1, x2 ∈ R, we have(a) ax1ax2 = ax1+x2 ;

(b)ax1

ax2= ax1−x2 ; and

(c) (ax1)x2 = ax1x2 .(d) Furthermore, if a �= 1, then ax1 = ax2 if and only if x1 = x2.

Example 4.2.1. The graph of the exponential function y = 2x is shown below:

y = 2x

Example 4.2.2. The graph of the exponential function y = (1/3)x is shown below:

y =(

13

)x

Example 4.2.3. We have 8x × 24x = (23)x × 24x = 23x24x = 27x = (27)x = 128x.

Example 4.2.4. We have 9x/2 × 27x/3 = (91/2)x × (271/3)x = 3x3x = 32x = (32)x = 9x.



32x+2 ÷ 82x−1 = 32x+2 × 81−2x = (25)x+2 × (23)1−2x = 25x+1023−6x = 213−x.


642x ÷ 162x = 212x ÷ 28x =212x

28x= 24x = (24)x = 16x.


5x+1 + 5x−1

5x+2 + 5x=

5x+1 + 5x−1

5 × 5x+1 + 5 × 5x−1=

5x+1 + 5x−1

5(5x+1 + 5x−1)=

15.


4x − 2x−1

2x − 12

=22x − 2−12x

2x − 12

=2x2x − 1

22x

2x − 12

=2x(2x − 1

2 )2x − 1

2

= 2x.

Example 4.2.9. Suppose that

25x =1√125

.

We can write 25x = (52)x = 52x and

1√125

= (√

125)−1 = ((125)1/2)−1 = ((53)1/2)−1 = 5−3/2.

It follows that we must have 2x = −3/2, so that x = −3/4.

Example 4.2.10. Suppose that (19

)2x−1

= 3(27−x).

We can write 3(27−x) = 3((33)−x) = 3 × 3−3x = 31−3x and(19

)2x−1

=(

132

)2x−1

= (3−2)2x−1 = 32−4x.

It follows that we must have 1 − 3x = 2 − 4x, so that x = 1.

Example 4.2.11. Suppose that 9x =√

3. We can write 9x = (32)x = 32x and√

3 = 31/2. It followsthat we must have 2x = 1/2, so that x = 1/4.

Example 4.2.12. Suppose that 53x−4 = 1. We can write 1 = 50. It follows that we must have3x − 4 = 0, so that x = 4/3.

Example 4.2.13. Suppose that (0.125)x =√

0.5. We can write

(0.125)x =(

18

)x

=(

123

)x

= (2−3)x = 2−3x

and√

0.5 = (0.5)1/2 =(

12

)1/2

= (2−1)1/2 = 2−1/2.

It follows that we must have −3x = −1/2, so that x = 1/6.

y

x

1 e

1


Example 4.2.14. Suppose that 81−x × 2x−3 = 4. We can write 4 = 22 and

81−x × 2x−3 = (23)1−x × 2x−3 = 23−3x × 2x−3 = 2−2x.

It follows that we must have −2x = 2, so that x = −1.

4.3. The Logarithmic Functions

The logarithmic functions are the inverses of the exponential functions. Suppose that a ∈ R is a positivereal number and a �= 1. If y = ax, then x is called the logarithm of y to the base a, denoted by x = loga y.In other words, we have

y = ax if and only if x = loga y.

For the special case when a = 10, we have the common logarithm log10 y of y. Another special case iswhen a = e = 2.7182818 . . . , an irrational number. We have the natural logarithm loge y of y, sometimesalso denoted by log y or ln y. Whenever we mention a logarithmic function without specifying its base,we shall assume that it is base e.

Remark. The choice of the number e is made to ensure that the derivative of the function ex is alsoequal to ex for every x ∈ R. This is not the case for any other non-zero function, apart from constantmultiples of the function ex.

Example 4.3.1. The graph of the logarithmic function x = log y is shown below:

x = log y

LAWS FOR LOGARITHMIC FUNCTIONS. Suppose that a ∈ R is positive and a �= 1. Thenloga 1 = 0 and loga a = 1. For every positive y1, y2 ∈ R, we have(a) loga(y1y2) = loga y1 + loga y2;

(b) loga

(y1

y2

)= loga y1 − loga y2; and

(c) loga(yk1 ) = k loga y1 for every k ∈ R.

(d) Furthermore, loga y1 = loga y2 if and only if y1 = y2.

INVERSE LAWS. Suppose that a ∈ R is positive and a �= 1.(a) For every real number x ∈ R, we have loga(ax) = x.(b) For every positive real number y ∈ R, we have aloga y = y.


Example 4.3.2. We have log2 8 = x if and only if 2x = 8, so that x = 3. Alternatively, we can useone of the Inverse laws to obtain log2 8 = log2(23) = 3.

Example 4.3.3. We have log5 125 = x if and only if 5x = 125, so that x = 3.




log4

18

= x if and only if 4x =18.

This means that 22x = 2−3, so that x = −3/2.


log2

√2

16= log2

(21/2

24

)= log2(2

−7/2) = −72.

Example 4.3.8. We havelog3

13√

3= log3(3

−3/2) = −32.

Example 4.3.9. Use your calculator to confirm that the following are correct to 3 decimal places:

log10 20 ≈ 1.301, log10 7 ≈ 0.845, log10

15≈ −0.698,

log 5 ≈ 1.609, log 21 ≈ 3.044, log 0.2 ≈ −1.609.


loga

13

= −13

if and only if a−1/3 =13.

It follows that

a = (a−1/3)−3 =(

13

)−3

= 33 = 27.


loga

14

= −23

if and only if a−2/3 =14.

It follows that

a = (a−2/3)−3/2 =(

14

)−3/2

= 43/2 = 8.


loga 4 =12

if and only if a1/2 = 4.

It follows that a = 16.


Example 4.3.13. Suppose that log5 y = −2. Then y = 5−2 = 1/25.

Example 4.3.14. Suppose that loga y = loga 3 + loga 5. Then since loga 3 + loga 5 = loga 15, we musthave y = 15.

Example 4.3.15. Suppose that loga y + 2 loga 4 = loga 20. Then

loga y = loga 20 − 2 loga 4 = loga 20 − loga(42) = loga 20 − loga 16 = loga

2016

,

so that y = 20/16 = 5/4.

Example 4.3.16. Suppose that12

log 6 − log y = log 12.

Then

log y =12

log 6 − log 12 = log(√

6) − log 12 = log√

612

,

so that y =√

6/12.

For the next four examples, u and v are positive real numbers.

Example 4.3.17. We have loga(u10) ÷ loga u = 10 loga u ÷ loga u = 10.

Example 4.3.18. We have 5 loga u − loga(u5) = 5 loga u − 5 loga u = 0.


2 loga u + 2 loga v − loga((uv)2) = loga(u2) + loga(v2) − loga(u2v2) = loga(u2v2) − loga(u2v2) = 0.

Example 4.3.20. We have(loga

u3

v+ loga

v

u

)÷ loga(

√u) = loga

(u3

v× v

u

)÷ loga(u1/2)

= loga(u2) ÷ loga(u1/2) = 2 loga u ÷ 12

loga u = 4.

Example 4.3.21. Suppose that 2 log y = log(4 − 3y). Then since 2 log y = log(y2), we must havey2 = 4 − 3y, so that y2 + 3y − 4 = 0. This quadratic equation has roots

y =−3 ±

√9 + 16

2= 1 or − 4.

However, we have to discard the solution y = −4, since log(−4) is not defined. The only solution istherefore y = 1.

Example 4.3.22. Suppose that log(√

y) =√

log y. Then

12

log y =√

log y, and so log y = 2√

log y.

Squaring both sides and letting x = log y, we obtain the quadratic equation x2 = 4x, with solutionsx = 0 and x = 4. The equation log y = 0 corresponds to y = 1. The equation log y = 4 corresponds toy = e4.


Problems for Chapter 4

1. Simplify each of the following expressions:a) 2162/3 b) 323/5 c) 64−1/6 d) 10000−3/4

2. Simplify each of the following expressions, where x denotes a positive real number:

a) (5x2)3 ÷ (25x−4)1/2 b) (32x2)2/5 ×(

25x4

)1/2

c) (16x12)3/4 ×(

27x6

)−1/3

d) (128x14)−1/7 ÷(

19x−1

)−1/2

e)(2x3)1/2

(4x)2÷ (8x)1/2

(3x2)3

3. Simplify each of the following expressions, where x and y denote suitable positive real numbers:

a) (x2y3)1/3(x3y2)−1/2 b)x−1 − y−1

x−3 − y−3

c)x−2 + y−2

x + y− x−2 − y−2

x − yd) xy ÷ (x−1 + y−1)−1

e) (x−2 − y−2)(x − y)−1

(1xy

)−1

(x−1 + y−1)−1 f) (36x1/2y2)3/2 ÷(

6x−2

y−2/3

)3

4. Determine whether each of the following statements is correct:a)

√8 + 2

√15 =

√3 +

√5 b)

√16 − 4

√15 =

√6 −

√10

5. Simplify each of the following expressions, where x denotes a positive real number:

a) 32x × 23x b) 163x/4 ÷ 42x c)2x+1 + 4x

2x−1 + 1d)

9x − 4x

3x + 2x

6. Solve each of the following equations:

a) 272−x = 9x−2 b) 4x =1√32

c)(

19

)x

= 3 × 81−x

d) 2x × 16x = 4 × 8x e) 4x = 3x f)(

17

)x

= 72x

7. Find the precise value of each of the following expressions:

a) log3

127

b) log100 1000 c) log5

√125 d) log4

132

8. Solve each of the following equations:a) log2 y + 3 log2(2y) = 3 b) log3 y = −3 c) log 7 − 2 log y = 2 log 49d) 2 log y = log(y + 2) e) 2 log y = log(7y − 12) f) log y = 3

√log y

g) 2 log y = log(y + 6) h) 2 log y =√

log y

9. Simplify each of the following expressions, where u, v and w denote suitable positive real numbers:

a) logu4

v2− log

v2

ub) log(u3v−6) − 3 log

u

v2

c) 3 log u − log(uv)3 + 3 log(vw) d) loguv

w+ log

uw

v+ log

vw

u− log(uvw)

e)(

logu4

v2− log v + 2 log u

)÷ (2 log u − log v) f) log(u2 − v2) − log(u − v) − log(u + v)

g) log u3 + log v2 − log(u

v

)3

− 5 log v h) 2 log(eu) + log(ev) − log(eu+v)

− ∗ − ∗ − ∗ − ∗ − ∗ −

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